Dispersive Qubit Readout — Circuit QED Measurement

Qubit State

Excited-state population p₁ 50%

Readout Chain

Coupling g / 2π 60 MHz

Drive amplitude ε 2.5

Integration time τ 500 ns

Live Readout

Dispersive shift χ/2π

3.6 MHz

Pointer separation D

0.94

Signal-to-noise (SNR)

2.2

Single-shot fidelity

92%

Assignment error (shots fired)

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How it works

A transmon qubit dispersively coupled (detuning Δ ≫ g) to a readout resonator shifts the resonator's frequency depending on its own state — the Jaynes–Cummings Hamiltonian in the dispersive limit reduces to:

H/ħ ≈ ω_r a†a + (ω_q/2)σ_z + χ σ_z a†a,   χ = g²/Δ

Driving the resonator at its bare frequency ω_r populates it with a coherent state whose steady-state amplitude depends on the qubit state, because the qubit pulls the resonator frequency by ±χ:

α_g = ε / (κ/2 − iχ)
α_e = ε / (κ/2 + iχ)
D  = |α_e − α_g| = 2εχ / [(κ/2)² + χ²]

The two pointer states α_g and α_e land at different points on the IQ (in-phase/quadrature) plane after being amplified and demodulated. Averaging the homodyne signal for a time τ suppresses noise as 1/√τ, so the effective signal-to-noise ratio and optimal single-shot discrimination fidelity are:

SNR(τ) = D·√τ / σ₀
F = ½[1 + erf(SNR / 2√2)]

This is exactly the interface real quantum processors use to talk to the classical world: every measurement on an IBM or Google transmon chip is this dispersive-shift trick, read out through an impedance-matched amplifier chain and turned into a 0/1 bit by a threshold on the IQ plane.

p₁ slider / Prepare buttons — sets the qubit's population before the readout pulse; only population (not superposition phase) is measured by a QND dispersive readout.
g, ε, τ — coupling strength, drive amplitude and integration time; watch how each reshapes the pointer separation and shrinks the noise clouds.
Fire 400 single shots — samples a run of individual measurement outcomes from the two Gaussian clouds and reports the fraction that would be assigned to the wrong state by an optimal threshold.

Qubit Bloch needle Readout resonator |g⟩ shots |e⟩ shots

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